Plot hole: Dr. Adam Soong is initially presented as a discredited scientist, banned from the scientific community; he gets debarred and his funding revoked. And it's not an internal matter; he is publicly exposed for it. His daughter in episode 6 even finds out this information on Google. Several news articles call him "mad scientist" and such. However, this same person at the same time throughout the rest of the season has every bit of pull and influence, not just through undercover channels, but is treated with the utmost honor and deference by the NASA PR people at public events.
Other mistake: The show is supposed to take place in the same timeline as Next Generation, but the presence of a young Guinan in 2024 who has never met Picard before contradicts the TNG Season 5 finale, Time's Arrow, where Picard met Guinan (looking like Whoopi Goldberg, who also appears here) in the XIX century. When asked about this contradiction, show writer Terry Matalas said that the Federation actually never happened since the 'future' was changed by Q's actions, so Picard never traveled back in TNG meeting her. However, that is never referenced in the actual episode or the season finale, and would create other paradoxes, especially since in the very same episode there is a reference to Star Trek IV. The same punk from back then rubs nervously his neck remembering Spock's nerve pinch; the TOS cast would have not visited XX century Earth in the way shown in the movie if the Federation never came into existence.
Plot hole: In this episode, the 'Watcher' displays the power to possess/mind control people at will, jumping from body to body to set up the meeting with Picard. Forgetting the fact that this usage of power for such a menial task is actually detrimental to what she wants to do (it leaves more evidence, by unnecessarily messing up with minds just to tell the guy to go from A to B), this would have been super-useful for the rest of the season, but she just never ever uses it again, not even a nerfed version of it.